High Energy Physics - Theory

We compute torus partition functions of orbifolds of two-dimensional chiral fermions by huge non-Abelian groups involving the Conway group, one of the sporadic finite simple groups. This seemingly standard problem turns out to contain, somewhat unexpectedly, multiple major issues hitherto unresolved, both at the level of the principle and at the level of practice. We describe in detail how they can be overcome. Although the methods described in this paper are generally applicable, the choice of the particular orbifolds we consider in this paper was motivated by the search for the periodicity element of topological modular forms. We will provide a brief introduction to this question, to explain why all the techniques in this paper had to be developed. Unfortunately, the orbifolds we considered turned out not to be the periodicity element.

The high order behavior of perturbation theory provides insight into non-perturbative phenomenon in quantum field theory. While extensively studied in simplified models, high order behavior is challenging to access in more realistic field theories due to the complexity of the standard Feynman diagrammatic expansion. Building on recent progress in global sampling on the moduli space of tropical curves, we introduce an annealed importance sampling framework for tropical field theory. Our approach overcomes the exponential weight-variance barrier, allowing precise numerical calculations to high orders. We apply our technique to compute the perturbative series for the two-point function in the massive three-dimensional $\phi^3$ theory through 325 loops. These coefficients determine the instanton action in $\phi^3$ theory to 0.03\% precision, confirm the collective-coordinate exponent, and extract the one-loop determinant and the instanton fluctuation corrections through four loops, in remarkable agreement with direct semiclassical calculations. Our approach is applicable to a variety of related field theories, facilitating the quantitative exploration of their high loop behavior.

The family of $SL(2,\mathbb Z)$ black holes has an accumulation point corresponding to a cusped geometry, which causes the sum over black holes to diverge. We extend cusp regularization to the sum over geometries compatible with a fixed boundary spin structure. For $c_L=c_R$, including the cusp as a counterterm cancels the leading divergence for each of the three even boundary spin structures. The required subtraction is the same in all three sectors and is two thirds of the bosonic one. We further show that gravitational anomalies cause cancellations between black hole contributions, removing the leading cusp divergence except at $c_L-c_R=0,\pm24$ in pure gravity and $c_L-c_R=0,\pm12$ in $\mathcal N=(1,1)$ supergravity. We also derive the spin structure dependent anomaly phases responsible for these cancellations.

We solve the Schrödinger equation for a particle in the circle with a quantum mechanical bootstrap method. The main idea is that in physical situations one needs solutions of the recursion relations for the Fourier coefficients of the measure that vanish exponentially fast. We show that this results in a (generalized) continued fraction representation of the solution. The solution is an analytical function of the couplings and the energy. We also show how one this method directly computes the density of states and checking that the answer is physical is reduced to checking an inequality for a single variable. We also show how to reconstruct the wave function and how to extract the quasimomentum of the particle as a function of the energy. We illustrate our solution in the simplest case of the Mathieu equation, where we show that the method can easily resolve exponentially small bands in the band spectrum.

We analyze the structure of the baby universe which arises in the coupled SYK realization of the AS$^2$ geometry. To do this, we develop and leverage combinatorial arguments about traces of many Majoranas which remain valid at finite $N$ and $p$ and thus furnish the finite $\lambda$ completion of the chord rules found in [1]. We find that there is a first order phase transition which persists in the $\beta\rightarrow\infty$ limit between the thermal AdS and AS$^2$ phases parameterized by the mass of the heavy particle insertion and the ratio between the SYK and Maldacena-Qi coupling constants. We then explore Lorentzian time evolution in the baby universe and find a cosine scale factor confirming the big bang / big crunch nature of the bulk cosmology.

The BPS states in Calabi-Yau compactifications of type II string theory can sometimes be described using quiver quantum mechanics whose Higgs branch is a complete intersection manifold. Their degeneracy resembles that of BPS black holes. A small fraction of these states corresponds to multi-center supergravity brane configurations and can be continued to the (near-Higgs) Coulomb branch. We identify them as monotone states. The overwhelming majority of states, known as pure-Higgs states, are fortuitous states, which concentrate at a particular $R$-charge. This division has a precise mathematical meaning as two different types of cohomologies of the Higgs branch target space. We argue that a natural probe of BPS chaos in these models is the Berry/Chern curvature. In the case of the Gepner model, we use localization techniques in an auxiliary two-dimensional CFT to compute the curvature, its RMT properties, and deviations from it.

We show that massless theories in odd-dimensional de Sitter space have very simple measures---the modulus squared of the late-time wavefunctional---at tree level. They are local functionals; in particular, all wavefunction coefficients vanish at separated positions. All the usual singularities of nonlocal wavefunctions in even dimensions only affect the phase in the odd dimensional case. Hence a closed formula can be written for any number of points. We focus on three examples: the non-linear sigma model, Yang-Mills theory and General Relativity, from dimensions three to seven.

We consider the 1/4-BPS operators in the D1-D5 system on AdS$_3\times S^3\times T^4$. We evaluate the anomalous dimensions of low-lying sectors in conformal perturbation theory around Sym$^N(T^4)$ exactly in $N$. All anomalous dimensions are consistent with recombination of the short multiplets into long ones and take a particularly simple form. We show that at finite $N$ and at first nontrivial order in the deformation, for the sectors we consider, the BPS density of states agrees with that of the supergravity point. To the order in conformal perturbation theory we consider, this confirms that not all states that can get lifted do get lifted.

We investigate axionic quintessence in heterotic string compactifications. Starting from the ten-dimensional theory, we derive the four-dimensional axion effective action, including gauge couplings and non-perturbative potentials, and identify the curvature terms that can correct it. We examine the geometric requirements for ultralight axions generated by exponentially suppressed worldsheet instantons, finding significant restrictions through an exploratory scan of toric Calabi-Yau hypersurfaces. In an illustrative K3-fibred setup, the minimal benchmark also encounters a tension with the positivity of the one-loop corrected gauge kinetic functions. Aligned worldsheet-instanton contributions provide a way to alleviate this tension and yield an ultralight axion with a phenomenologically relevant decay constant. Our analysis also identifies lower-action instantons whose contributions must be controlled to preserve the ultralight axion. We then study the resulting class of quintessence potentials and qualitatively compare their late-time cosmological solutions with observationally preferred ranges.

We investigate a non-Hermitian Schwinger effect by analyzing a one-loop effective Lagrangian of scalar and Dirac quantum electrodynamics (QED) in the presence of a real electric field and a parallel imaginary magnetic field. We show that an imaginary magnetic field produces an additional pole family in the integrand of the proper-time representation. Unlike a real magnetic field, an imaginary magnetic field can generate a nonzero imaginary part of the effective Lagrangian even in the absence of an electric field. The interplay between the two fields leads to nontrivial phenomena, such as a cancellation mechanism between the contributions of two fields. We also point out that when the ratio of two field strengths is rational, some poles among the two families coincide, and a careful treatment of double poles is required. By further examining the complex Landau-level structure of the electric-pole contribution, we find that the field dependence of the vacuum decay rate from each Landau level exhibits a sinusoidal-like behavior, and their sum determines the overall field dependence. Our results clarify the analytic structure of the QED effective Lagrangian in imaginary magnetic backgrounds and provide a basis for investigating its relation to non-Hermitian phenomena in the QED vacuum.

We ask when tree-level supersymmetric cohomology admits a Hironaka decomposition: a unique expansion in finitely many secondary operators with polynomial coefficients in algebraically independent bosonic primaries. Such a decomposition organizes the cohomology into towers with a common set of unrestricted dressings. For diagonal models with $N$ labels, $d\geq1$ bosonic species and $f$ fermionic species, we construct this decomposition and obtain $(N!)^{d-1}2^{fN}$ secondaries. We also prove that the scalar-branch ring of ABJM theory admits a Hironaka decomposition at every rank and positive integer Chern--Simons level, and construct explicit bases for the $SU(2)\times SU(2)$ Klebanov--Witten scalar ring. We also do the same for the Higgs branches of two class-S theories. However, not all supersymmetric cohomologies admit a Hironaka decomposition. In the Wess--Zumino model with $W=x^2y^2$, every positive-degree scalar annihilates the nonzero class $xy$, while $x^n$ is in cohomology for every $n\geq 1$. These two properties rule out a Hironaka decomposition over scalar primaries.

Modeling the gravitational waves from merging black holes requires precise calculations of the two-body problem. The post-Minkowskian expansion of black-hole scattering, in powers of Newton's constant, now stands at fifth order, four loops. At this order one part of the deflection diverges when the black holes pass at about 94 percent of the speed of light. The divergence is known to come from periods of a K3 surface, which has a node at this speed. We show that the node lies on the integration contour of the K3 maximal cut, which makes this speed a Landau singularity of the cut. Requiring the divergence to cancel, without introducing new infinities, has been shown to fix the memory contribution, named for the permanent displacement that the waves leave in test masses, up to one constant. We derive the leading coefficients of its boundary integrals in closed form. We obtain both results from maximal cuts and boundary limits, without an integration-by-parts reduction of the full four-loop integrand. We also continue the K3 and Calabi--Yau periods of these integrals to the energies of bound orbits, as in the inspiral before a merger.

These lectures provide an introduction to modern techniques for studying the analytic properties of loop-level scattering amplitudes and correlation functions in perturbative quantum field theory. We begin by developing the ideas behind Landau analysis, which identifies the location and nature of kinematic singularities in Feynman integrals (the universal building blocks that contribute to perturbative amplitudes and correlators). We then show how fundamental principles such as unitarity and causality constrain the analytic structure of Feynman integrals, and develop methods for working out some of these constraints in practice. Finally, we connect these observations to the properties of iterated integrals and their symbol calculus, to show how bootstrap methods can be used to determine the functional form of Feynman integrals from just knowledge of their analytic structure. These notes are based on a series of lectures held during the La Ricotta Summer School "The Disordered Universe 2025" from 21~July to 1~August~2025.

We develop a quantum hydrodynamic theory based on Nambu's unpublished note \emph{A New Look at Fluid Dynamics}. The kinetic action defines a metric on configuration space and a mechanical connection coupling an observable cyclic deformation to unobserved fluid modes. Tracing out these modes yields a quantum channel for the observed deformation. We establish a criterion for the exact preservation of coherence between opposite momenta and show how nonzero connection curvature prevents this coupling from being removed. The leading quadratic decay of quantum coherence is determined by the prepared quantum state. For two shear flows, we identify an exact noiseless sign subsystem. Introducing additional generating flows cancels the initial residual of the Euler equations while producing nonzero connection curvature, governed by the same kinetic integrals that determine the classical correction. At fixed mean energy, choosing amplitudes adapted to the dominant coupling flow substantially reduces the initial decoherence, even along the direction that cancels the Euler residual. This advantage gradually diminishes under conditional evolution, eventually leading to a crossing of the decoherence curves. Our example shows that a preparation with a smaller initial coherence-decay coefficient can have a larger maximum sampled loss over a finite readout interval.

In general trace relations appear in finite $N$ matrix models and collective field theories. They lead, as was established previously, to reductions of large $N$ Hilbert spaces. Even more importantly they induce the appearance of a whole class of new (miraculous) states whose number grows exponentially in $N^2$. In this work we discuss their relevance to finite $N$ thermodynamics and the role they play in constraining the partition functions of finite $N$ theories.

We study the role of the Hull connection in the heterotic superpotential for N=1 d=4 vacua. The integrand is defined in terms of H, which is in turn defined in terms of a gravitational Chern--Simons coupling involving H. We show this recursive structure is key to W reproducing some of the ten--dimensional supersymmetry conditions written down by Bergshoeff--de Roo through \alpha'^3. With this evidence in hand we make some observations about extending some of the tensor equations of Hull and Strominger to all orders in \alpha'.

The complete Wess-Zumino (WZ) action and stress-energy tensor (SET) of the conformal anomaly in both two and four dimensions is given. This local effective action is expressed in terms of a single scalar Nambu-Goldstone `conformalon' field $\varphi$, parametrizing the local abelian Weyl group of conformal transformations that act transitively on the spacetime metric. Satisfying WZ consistency, and equipped with a variational principle for $\varphi$ that determines the conformal transformation $g_{\mu\nu}\!\to\!g_{\mu\nu}e^{-\varphi}$ to an associated spacetime with no anomaly, the WZ action resolves the conformal gauge fixing ambiguity of earlier methods of `integrating the anomaly.' In $D\!=\!2$ the WZ action is the local form of the Polyakov action, whose massless $\varphi$ propagator describes a correlated two-particle scalar intermediate state in the SET polarization function $\langle TT\rangle$. The imaginary part of this polarization obeys a UV finite sum rule, becoming a Dirac $\delta$-function at zero energy for conformal quantum fields, demonstrating the infrared origin of the anomaly, while the real part becomes the light cone propagator of $\varphi$. Although the $\Box R$ term in $D\!=\!4$ corresponds to a cohomologically trivial cocycle, it also arises from the IR limit of massless quantum fields, independently of UV physics, and contributes to the WZ action for low energy gravity. Analogous to the WZ action for pseudo-Goldstone mesons in low energy QCD from the axial anomaly, the WZ action for the conformal anomaly provides a Nambu-Goldstone scalar field for macroscopic gravity, in addition to the metric of classical general relativity.

Extending our previous work, we investigate Wilson loops of flavored Chern-Simons theory and ABJ theory by the use of Picard-Fuchs equations. The singularity structure of the equations is analyzed in detail, and local solutions around the singularities are determined. Each theory has one parameter, and the parameter dependence of the Wilson loops is also determined. In flavored Chern-Simons theory, we find an interesting phase transition. For $n$-times-wound Wilson loops, we determine their $n$-dependence by using a recursion relation obtained in our previous work.

Effective field theories (EFTs) offer a powerful method to unify diverse models under a small set of control parameters, allowing systematic expansions around well-established theories. While these techniques were traditionally developed for particle physics experiments with clean, isolated initial states, realistic macroscopic environments are often noisy and dissipative. Recognising the limitations of traditional EFTs has recently sparked intense progress at the interface of high-energy physics and condensed matter. These developments motivate a new approach to gravitation and cosmology, one that models the gravitational sector as evolving in the presence of an unobservable medium. Open Effective Field Theories provide a systematic, controllable framework for this endeavour. This chapter presents this versatile framework and illustrates how the Schwinger-Keldysh formalism captures the influence of unknown environments, manifesting as dissipation and noise, on observational probes. These notes are based on a series of lectures held during the La Ricotta Summer School "The Disordered Universe 2025" from 21 July to 1 August 2025.

These lecture notes provide an introduction to quantum spin glasses, with a particular emphasis on their application to open problems in cosmology and holography. Focusing on the spherical $p$-spin model, we develop a complete and self-contained analysis of its thermodynamic and dynamical properties. We introduce the replica method, discuss both replica-symmetric and one-step replica symmetry-broken solutions, and characterise the resulting glassy phase. The framework is then extended to the quantum model, where we study the emergence of slow dynamics and the role of quantum correlations through Euclidean and real-time correlation functions. Finally, we explore how concepts such as replica symmetry breaking and disorder naturally arise in cosmological and holographic systems related to black hole fragmentation, highlighting spin glass models as useful theoretical laboratories for these phenomena. These notes are based on a series of lectures held during the La Ricotta Summer School: The Disordered Universe 2025 from 21 July to 1 August 2025.

In theories with compact extra dimensions, the four-dimensional effective description is obtained by Kaluza-Klein reduction. The state available to a four-dimensional observer is the reduced density matrix obtained by tracing the higher-dimensional vacuum over the tower of massive modes, and it need not be pure. In this paper we study the momentum-space entanglement of this state for a five-dimensional scalar on $M_4\times S^1$ with a cubic or quartic self-interaction. We compute, at leading order in the coupling, the mutual information $I_n$ between the zero modes below a resolution scale $\mu$ and a single Kaluza-Klein level $n$. When the zero modes couple linearly and without derivatives to pairs of tower modes, $I_n$ falls as $1/m_n$, where $m_n$ is the mass of level $n$. In the cubic theory, this tail is independent of the ultraviolet cutoff, with coefficient $(\mu R)^2/1024\pi^3$ for compactification radius $R$, and the mutual information with the whole tower grows logarithmically with the number of levels. In the quartic theory, the tail is masked by states containing additional hard quanta, which make $I_n$ depend on the cutoff. Compared with the corrections the tower makes to low-energy amplitudes, which are dominated by the lightest levels and captured by local operators, the mutual information with the tower is not dominated by any finite set of levels and arises from the time-nonlocal part of the effective action. The local four-dimensional effective theory therefore describes the low-energy observables of the reduced system but not its state. Unbroken supersymmetry cancels the vacuum energy of the compactification but not the mutual information, which at leading order is a sum of squared amplitudes.

We compute the normal mode spectrum and its spectral form factor for a probe scalar wave in smooth horizonless backgrounds. We first consider the BTZ black hole with a brickwall that hides the near-horizon region, which has recently been proposed as a toy model for a finite-temperature semiclassical black hole microstate. In this setup, the spectral form factor may or may not have a linear ramp, depending on the details of the brickwall construction. However, we argue that even in cases that do show a ramp, this ramp does not originate from level correlations associated with quantum chaos. Then we calculate the normal mode spectrum for a genuine microstate geometry (superstratum) and find evidence of chaos, both with a ramp and level repulsion. We conclude that the eigenvalue spectrum of normal modes can indeed serve as a tool for investigating the quantum dynamics of fuzzballs; however, the brickwall models are incapable of capturing this phenomenon.

We introduce a simple functional of algebra-valued fields with definite Lorentz properties that is kernel-free yet it encodes the structure of tree-level scattering amplitudes. The fields are evaluated at the points of a piece-wise straight paths in the anti-self-dual plane of complexified Minkowski space, with the directions of consecutive segments correlated so that the path has a velocity memory. We show that momentum-space reduction of the functional generates the cyclic Parke--Taylor factor, off shell in terms of light-front bilinears and on shell in terms of spinor brackets. The same reduction, organized differently, can be naturally promoted to twistor space without any twistor-space input: a closed cyclic chain of Penrose transforms on a twistor line, in which the Parke--Taylor denominators arise as boundary terms of the integrals along the paths. Finally we show that derivatives with respect to one of the path parameter generate numerators that together with the Parke-Taylor denominator build scattering amplitudes. Lorentz invariance fixes the spins of the inserted fields and the homogeneities of the corresponding twistor functions without any use of supersymmetry. The three-point outputs reproduce the kinematic forms of the light-cone cubic vertices of chiral higher-spin theory, and the derivative monomials of order four generate the components of the $\overline{\rm MHV}$ supermultiplet of $\mathcal N=4$ super Yang--Mills theory, off shell and on shell.

We revisit the 2-loop computation of the partition function of the GS superstring expanded near the AdS$_2$ surface in AdS$_5\times S^5$ and AdS$_4\times {\rm CP}^3$, dual to $\frac{1}{2}$ BPS circular Wilson loop in ${\cal N}=4$ SYM and ABJM theories. In arXiv:2601.08809 it was done in AdS$_5\times S^5$ using dimensional regularization and a UV pole was found. Since dimensional regularization is problematic for the GS string, here we use a heat-kernel cutoff instead. Then the contribution of the local path integral measure, proportional to $\delta(0)$, can no longer be ignored. We show that it cancels the $\delta(0)$ terms coming from the quartic interactions in the action, leaving the square of the difference ${\rm Str}\,E$ between the regularized numbers of bosonic and fermionic modes. The result is thus UV finite, the pole of arXiv:2601.08809 being an artefact of dimensional regularization. If all fields are regularized with the same smearing kernel, ${\rm Str}\,E=0$ and the 2-loop coefficient $f_2$ vanishes, in tension with $f_2=\frac{3}{32\pi^2}$ in SYM theory. In the AdS$_4\times{\rm CP}^3$ case the same cancellation leaves a finite term proportional to the square of the coincident value of the massless 2d fermion Green's function, giving, ironically, $f_2=\frac{3}{32\pi^2}$ instead of $f_2=0$ required by localization. This term is the level-zero (string) part of the result of arXiv:2609.14497 for the AdS$_2\times S^1$ M2 brane in AdS$_4\times S^7/\mathbb Z_k$, where it is cancelled by the higher $S^1$ modes, which also remove a finite mass shift of the massless fermion in conflict with the AdS$_2$ supersymmetry. This suggests that the type II string path integral measure should contain an extra contribution, supplied in the IIA case by the parent M2 brane; in the AdS$_5\times S^5$ case it is still to be identified and may resolve the remaining 2-loop mismatch with localization.

A complete understanding of cosmology, from its earliest epoch all the way to the current dark energy dominated era, requires the development of a mathematical framework that incorporates quantum features in an expanding universe. Such a framework, alongside a description of the underlying degrees of freedom, is so far lacking. Several of the pertinent challenges and puzzles already arise upon considering even the most idealised model of an expanding universe --- the de Sitter spacetime and fluctuations thereof. In these lectures, we review some elementary ideas of the de Sitter physics, passing through the representation theory of the de Sitter group, aspects of canonical quantum gravity, all the way to attempts in describing quasi-local features. Throughout, we highlight open problems that motivate ongoing efforts to understand quantum gravity and cosmology in universes with $\Lambda>0$. These notes are based on a series of lectures held during the La Ricotta Summer School: The Disordered Universe 2025 from 21 July to 1 August 2025.

The Disordered Universe is a novel research direction that exploits ideas and techniques coming from apparently unrelated research areas, ranging from cosmology to scattering amplitudes in particle physics and disordered complex systems, with the aim of addressing open questions in cosmology at all scales, from the very infra-red (or very late time) to the early universe. This paper acts as a general introduction and vision for a set of lectures delivered at the first La Ricotta Summer School "The Disordered Universe 2025", held from 21 July to 10 August 2025. These lectures involved open quantum systems, analytic properties of scattering amplitudes and correlation functions, foundations and quantum aspects of the de Sitter universe, as well as statistical physics out of equilibrium and spin glasses. While the lecture notes related to each course taught at the school will be published separately and are inevitably devoted to both conceptual and technical aspects of each of these topics, the school was not meant to be a collage of independent topics: the present paper has the purpose of outlining the global vision, how cosmology ends up being a unifier and benefits from both their viewpoint and techniques, how all the concepts and techniques originally coming from these four different areas are part of the same cosmological narrative, why this point of view deserves to be explored, and drawing perspectives for the future. We consider it crucial to build a microscopic theory for the early universe and maybe one that embraces all scales.

We construct stationary axially symmetric black holes with scalar hair in the gravitating nonlinear $O(3)$ sigma model with a partly negative symmetry-breaking potential. Our results show that it supports two qualitatively different types of asymptotically flat hairy black holes, distinguished by their connection to the corresponding solitonic branches and by the structure of their scalar hair; the type I black holes are connected to the vacuum branch and type II black holes are connected to regular scalaron-supported spinning solitons. The type II black holes have a strongly deformed scalar field structure, with the scalar field taking its vacuum value on the symmetry axis and approaching the anti-vacuum in the near-horizon equatorial region. We analyze their global charges, horizon properties, and dependence on the model parameters, comparing the solutions with Kerr black holes. For sufficiently weak gravitational coupling, the branch connectivity changes: the fundamental branches of types I and II reconnect to form one continuous family, while their corresponding second branches form another. We find that, in contrast to the vacuum-connected type I family, the scalaron-supported type II black holes extend to substantially larger horizon radii while preserving a nontrivial scalar configuration outside the horizon.

We derive explicit overlap functions for integrable matrix product states associated with arbitrary finite-dimensional irreducible evaluation representations of the orthogonal and symplectic twisted Yangians. The central step is to relate the commuting quasi-determinants entering the universal overlap formula to Sklyanin determinants of nested subalgebras. For orthogonal evaluation representations, this relation determines their spectra directly in the Gelfand--Tsetlin basis. In the symplectic case, we combine Sklyanin determinants and minors with a factorization of the nested rank-two boundary matrices to obtain the corresponding spectra. The resulting overlap functions are explicit products parametrized by Gelfand--Tsetlin patterns and require only the highest-weight data of the underlying orthogonal or symplectic representation. Together with the universal ratio of Gaudin determinants and the model-dependent pseudovacuum overlap coefficients, they determine the normalized MPS overlaps. We also express the overlap functions in the orthogonal case beyond evaluation representations in terms of the highest weights arising along the nested twisted-Yangian subalgebras.

Central extensions of asymptotic symmetry algebras are traditionally determined through an explicit computation of the algebra of conserved charges. We show that, within an ambiguity-free covariant phase space formalism based on the Noetherian split, they instead arise as intrinsic quantities completely determined by the Lagrangian through the geometry of the associated covariant phase space. We prove two general Charge Representation Theorems describing the canonical realization of diffeomorphism symmetries in the presence of field-dependent vector fields and Lagrangian anomalies, corresponding respectively to vanishing and integrable Noetherian flux. In both cases, we derive explicit expressions for the associated central extensions and establish the corresponding cocycle identities. This provides a constructive algorithm for determining central charges directly from the Lagrangian formulation of the theory, without requiring an explicit computation of the charge algebra. As a non-trivial application, we consider asymptotically locally AdS$_3$ spacetimes with Penrose--Brown--Henneaux boundary conditions. We recover the corresponding centrally extended asymptotic symmetry algebra, providing an explicit realization of the Charge Representation Theorems.

We propose bounds relating thermal entropy to entanglement in unitary conformal field theories: $f\le4\pi^2a$ in four dimensions and $f\le2.99830949\ldots\,F$ in three dimensions. Here $f$ is the dimensionless coefficient of the cosmological constant term in the thermal effective action, $a$ is the Euler anomaly coefficient, and $F$ is the sphere free energy. Spherical Rényi entropy relates the four-dimensional bound to the conformal collider bounds. We formulate a stronger conjectural inequality involving the stress-tensor three-point function, saturated by every free theory, and establish it in controlled regimes of weakly coupled gauge theories, large-$N$ vector models, and holographic CFTs. Finite-$N$ numerical results provide additional tests. The bounds constrain the Cardy growth of the density of states at high energy.

Recently, a general exact autonomy framework for $T\bar{T}$-like deformations of 2D integrable sigma models and 4D nonlinear electrodynamics has been developed. The theories are characterized by the Courant-Hilbert equation and admit an equivalent description in the Russo-Townsend auxiliary-field formulation. For general one-scale families, we derive an exact irrelevant flow equation of the universal form $\partial_\lambda \mathcal{L}=X\mathcal{F}(\sqrt{Y/X}), \,\,\,X=T_{\mu\nu}T^{\mu\nu}, \,\,\, Y=(T^\mu_\mu)^2$ where the function $\mathcal{F}$ is determined by the underlying Courant-Hilbert solution. We obtain a parametric representation valid for arbitrary one-scale models, establish its perturbative expansion in terms of the defining function or auxiliary potential, and identify the domain in which the flow remains regular. We also show that the irrelevant flow commutes with the marginal root-$T\bar{T}$ deformation and that both flows preserve classical integrability. Explicit results are derived for generalized Born-Infeld, q-deformed, no-$\tau$-maximum, and logarithmic models, with the latter admitting a closed representation. Finally, we extend the flow construction to arbitrary spacetime dimension $(d)$, revealing a common one-variable structure that connects two-dimensional integrable sigma models to four-dimensional self-dual nonlinear electrodynamics. These results provide a systematic framework for constructing and classifying exact $T\bar{T}$-like flows beyond the standard Born-Infeld setting.

The leading quantum correction to the Newton potential induced by the one-loop vacuum effective action of gravity is of order $1/r^3$. One-loop corrections are also responsible for the trace anomaly. We first show, following our previous work, that the effective action obtained by integrating the trace anomaly cannot reproduce this correction. We then complete the semiclassical Einstein equations with the vacuum-polarization stress tensor in addition to the anomalous stress tensor. The full stress tensor follows from the approximation proposed by Visser for the Schwarzschild spacetime. The stress tensor is taken in the Boulware vacuum and expanded at large distance, where it is conserved and reproduces the trace anomaly. Solving the equations perturbatively in $\hbar$, we find that Visser's approximation generates the $1/r^3$ correction to the Newton potential, linear in the source mass $M$, with the same structure as the one-loop result. Its coefficient is controlled by a single free parameter of Visser's polynomial ansatz, which can be tuned to match the one-loop value. The dependence on the trace-anomaly coefficient appears only at order $1/r^4$.

We construct a classical reflected phase space for open Carroll--Weyl null strings with Dirichlet conditions at the target origin or linear mixed target gluings. The reflection-compatible constraints are differentiable, their Hamiltonian flows preserve the boundary domain, and the constraint system is weakly first class on its regular part. The induced boundary algebra extends the boundary Carrollian conformal algebra by an even weight-one Carroll--Weyl current, with normalization fixed by the reflected symplectic form. The regular local embedding count agrees with that of the closed theory. We exhibit a four-dimensional mixed sector with one canonical field pair and a separate solution family carrying a nonzero conserved target-space boost charge and a nondegenerate symplectic pullback. Smooth fixed-point reflection selects characteristic endpoints. For timelike defects with natural tangential variations and no additional boundary interactions, non-characteristic endpoints have vanishing Carroll flow. For displaced Dirichlet data, we derive the canonical affine lift and its required shift of the edge momentum by integrated bulk momentum.

The momentum-space thermal OPE generically gives an asymptotic high-energy expansion of thermal two-point functions. We study the resurgent structure of this expansion in holographic CFTs at infinite and finite volume, whose dual bulk geometries are planar and spherical AdS black holes, respectively. The perturbative expansion is fixed to all orders by the stress-tensor sector of the thermal OPE and can be computed from a near-boundary analysis of the bulk wave equation. Non-perturbative effects, on the other hand, encode information about the bulk geometry away from the boundary. Using exact WKB analysis, we identify the locations of the associated Borel singularities as WKB actions between the AdS boundary and distinguished points in the complex radial plane, which include the black hole singularity in certain regimes. Furthermore, we identify a complex-frequency chamber in the timelike momentum regime where Borel resummation of the high-energy expansion reconstructs the full thermal two-point function. We further derive Stokes jumps that determine the non-perturbative contributions required upon crossing the boundary of this chamber.

We derive the radiation-reaction equations of motion for a spinning charged body, the $\sqrt{\text{Kerr}}$ particle, through quadratic order in its spin, generalising the Abraham--Lorentz--Dirac equation. Extending an argument of Coleman, we obtain radiation reaction equations by evaluating the self-field of the spinning body on its own worldline, and removing its divergences with dimensional regularisation. We use the resulting equations of motion to compute the electromagnetic waveform emitted in the scattering of the spinning body off a spectator charge, both at tree level and including the leading radiation-reaction correction, which enters at one loop. Then, we apply the construction introduced in a previous work by some of the authors -- where radiation reaction is encoded in unitarity cuts involving Compton amplitudes -- to spinning particles, and recompute the radiation-reaction contribution to the one-loop waveform. We find that the waveform calculated using classical and amplitude methods agree through quadratic order in spin, providing a check on our radiation-reaction equations. Along the way, we also reproduce known results for linear-in-spin self-force effects and for the spinless self-force in six dimensions.

We investigate whether a stochastic spin-2 gauge-field background can stabilize the chromomagnetic condensate of axion--SU(2) Yang--Mills theory at fixed axion Noether charge in flat spacetime. By coarse-graining the gauge field, we uncover a new class of statistically homogeneous and isotropic spin-2-dressed chromomagnetic condensates satisfying the classical background equations. We then study their fluctuations in two steps. First, at the mean-field (Hartree) level, we determine the fluctuation spectrum and the region where all modes remain positively gapped. Second, we include the leading scattering corrections from fluctuations propagating through the stochastic background. We find that gap survival is controlled by non-Gaussian correlations of the dressing. Statistical positivity constrains the otherwise undetermined higher-point contributions through model-independent consistency conditions, without fixing their detailed momentum dependence. These bounds separate the Hartree-gapped region into a part where a positive Born-level gap is ruled out and a part where stability remains possible. The results are relevant to non-Abelian gauge condensates in early-universe cosmology, including inflation, dark energy, and primordial gravitational waves, and to QCD-like gauge sectors above confinement where thermal effects are subleading.

We solve large-$N$ two-dimensional QCD with fundamental quarks---the 't Hooft model---on rigid AdS$_2$ and dS$_2$. Their isometries preserve the null gauge $A_-=0$ and in affine coordinates the curvature enters the light-front Hamiltonian only through the special-conformal generator of the null line. The $SL(2,\mathbb{R)}$ Casimir then results in an exact meson equation: 't Hooft's equation plus a Legendre operator whose sign is determined by the curvature. On AdS$_2$ it is related to equations proposed in the context of folded/segmented strings and light-front holography. The pion is massless in the chiral limit and obeys a Gell-Mann--Oakes--Renner relation that fixes the chiral condensate, the dimensions cross over from Regge to logarithmic growth once the flux tube exceeds the curvature radius, and the flat-space integrable structure survives. We also solve the problem of a quark interacting with an antiquark placed at the boundary: beyond a critical coupling the string pulls it back to the boundary, resulting in a scattering phase. On dS$_2$ and due to the expansion of space, the mesons are unstable resonances even at $N=\infty$ and eventually dissociate. Their widths are given by Boltzmann factors at the de Sitter temperature.

We investigate the non-relativistic limit of the generalized Dirac equation in a weak, static, and spherically symmetric background of symmetric teleparallel gravity. The underlying generalized spinor connection incorporates the complete Clifford-algebra basis and introduces additional couplings to the non-metricity sector beyond those of the conventional Dirac theory. Working in the coincident gauge and adopting the weak-field Schwarzschild geometry in isotropic coordinates, we derive the corresponding generalized Dirac Hamiltonian and perform successive Foldy--Wouthuysen transformations up to order $1/m^2$, retaining terms to first order in the gravitational potential and its spatial derivatives. The resulting block-diagonal Hamiltonian contains not only the expected gravitational counterparts of the kinetic, spin--orbit, and Darwin interactions, but also additional operator structures generated by the generalized spinor connection. In particular, direct spin--gravity, anisotropic spin--momentum--gravity, and tidal spin--momentum couplings arise naturally from the generalized metric-affine interaction. We further perform an order-of-magnitude analysis for an electron in the Earth's weak gravitational field to justify the adopted truncation of the inverse-mass expansion. These results demonstrate that the generalized Dirac equation in a symmetric teleparallel background gives rise to new low-energy interaction channels involving the fermion spin, momentum, and spatial derivatives of the gravitational field. The resulting effective Hamiltonian provides a framework for exploring phenomenological constraints on the additional couplings entering the generalized spinor connection.

Beyond math and code, Large language models (LLMs) are beginning to tackle problems that have traditionally required substantial training for theoretical physicists, but it remains unclear how far this capability extends and where it might break. As a first step toward pushing LLMs in high-energy theoretical physics, we introduce QFT-Bench, a benchmark of 75 graduate- to research-level problems requiring extensive reasoning and derivation across core Quantum Field Theory (QFT) subareas. Each question has a well-defined target and closed-form reference answer. Evaluation combines symbolic verification with LLM-based judge. Across 39 proprietary and open-source models, frontier models already demonstrate substantial capabilities on nontrivial QFT calculations, with the strongest models exceeding 95% accuracy. At the same time, clear gaps remain. Even strong models still make errors in calculations, conventions, and instruction following, while weaker models often fail at the level of physical concepts or derivations. We further monitor models' CoTs and show that current models exhibit nontrivial reasoning unfaithfulness, i.e., they could reach correct answers but the reasoning itself does not justify and logically support them. These results provide a quantitative assessment of current LLM capabilities in QFT and a first step toward broader evaluation in high-energy theoretical physics.

We formulate the closure problem for first-order, traceful radiation-gauge metric reconstruction on a Kerr background with matter. The reconstruction is decomposed into Weyl-curvature, Ricci-trace, and mass--angular-momentum completion sectors. For compatible initial, gauge, and completion data, we establish local existence and uniqueness of the resulting triangular transport system on a regular Kerr domain, and identify the additional conditions needed for a closed global formulation. This is not a global existence theorem. We test the framework using two conserved matter sources. For a thin shell around a Schwarzschild black hole, Israel matching, regularity, asymptotic flatness, a barotropic equation of state, and fixed rest mass determine the static, axial, and leading rotational sectors. The ADM-mass shift is $\sqrt{1-2M/R}$ times the shell rest mass, while the axial sector determines the angular momentum, moment of inertia, and interior frame dragging. We also construct a stationary constant-angular-momentum fluid torus on Kerr and compute its Newman--Penrose Ricci projections, showing that nonspherical sources generally excite coupled Ricci sectors. The shell provides a closed benchmark, while the torus illustrates the additional structure required for general Kerr sources.

String compactifications and runaway-dilaton models can contain light scalar directions. Motion along these directions changes gauge couplings, the QCD scale, quark masses, and Yukawa parameters. A given string-derived sector then predicts low-energy slopes. Its gauge kinetic functions, matter-metric or Yukawa slopes, source charge, scalar trajectory, and scalar range determine a Damour--Donoghue weak-equivalence-principle (WEP) vector and a clock-drift vector. The fit assumes one long-range scalar or an aligned multi-field direction. With that assumption, WEP tests and secular clock drifts probe related projections of the same scalar-slope vector. The minimal two-charge Damour--Donoghue fit uses Ti--Pt MICROSCOPE, Be--Ti torsion-balance, and $^{85}$Rb--$^{87}$Rb atom-interferometer data, then adds the Yb$^+$ E3/E2 secular $\alpha$-drift row. The WEP-only fit gives $D_{\hat m}=(-6.06\pm34.91)\times10^{-12}$ and $D_e=(8.80\pm46.46)\times10^{-12}$, with correlation $-0.99958$. The clock input is $\dot\alpha/\alpha=1.0(1.1)\times10^{-18}\,{\rm yr}^{-1}$. It matches the WEP marginal reach when $|\zeta|=2.37\times10^{-8}\,{\rm yr}^{-1}$, with $\zeta=\dot\varphi/\alpha_E$. For the benchmark $\dot\varphi=\xi H_0$ and $\alpha_E=10^{-3}\alpha_0$, this gives $\xi/\alpha_0\simeq0.34$. A model becomes testable after it supplies $r=D_{\hat m}/D_e$, the scalar amplitude, and $\zeta$. The Damour--Piazza--Veneziano runaway-dilaton estimate lies near $\eta\sim10^{-12}$ for order-unity $n=2$ parameters, well above the MICROSCOPE-level WEP scale. In the representative-isotope two-charge model, a MICROSCOPE-level Be--Ti-like row would reduce the long WEP axis by a factor $27.8$; a repeated Ti--Pt-like row would barely change it. The calculation tests compactification-level predictions for string-derived dilaton or modulus sectors after the model supplies its low-energy data.

We demonstrate the formation and subsequent evolution of Julia--Zee dyons in $(3+1)$-dimensional classical lattice simulations of an $\mathrm{SU}(2)\to\mathrm{U}(1)$ symmetry-breaking model in an expanding background. The gauge and Higgs fields are prepared thermally, subjected to an ultraviolet filter, and evolved through a mass quench and subsequent expansion. Classical dyons can carry continuous electric charge, and we find charge magnitudes exceeding four times the electric charge unit $g$. Their measured electric-charge magnitudes tend to decrease during the subsequent evolution, consistent with charge exchange with the surrounding fields. At late times, the magnetic monopoles persist while the number of dyons and their observed charge range decrease, with only a few identified dyons remaining at the end of the simulations.

We present AlgebrasTQD, an open-source code written in GAP (Groups, Algorithms, and Programming), for computing condensable algebras and the corresponding topological boundaries and interfaces of twisted quantum doubles $D^\omega(G)$. These algebras can be used to classify 1+1d gapped and gapless phases whose symmetry is related by gauging to a finite group $G$ with anomaly $\omega\in H^3(G,U(1))$, to study 2+1d anyon proliferation and to construct logical gates in non-Abelian topological quantum codes. The functions in AlgebrasTQD compute all condensable algebras $A(H,F,\gamma,\epsilon)$ with their algebra structure and anyons, sort them into isomorphism classes and identify the twin algebras that are inequivalent but share the same anyon decomposition. AlgebrasTQD also determines all the subalgebra relations, the corresponding Hasse diagram and the types of 1+1d phases for any chosen, possibly non-invertible, group-theoretical symmetry. For non-maximal algebras, the code computes the reduced topological order with its $3$-cocycle and the map of anyons across the topological interface. Finally, AlgebrasTQD includes functions that construct certain non-Clifford logical gates via non-Abelian surface codes. We summarize the formulas that are implemented in the AlgebrasTQD code, and we illustrate its use with Abelian and non-Abelian groups including $\mathbb{Z}_2^3$ with $\omega_\text{III}$, $D_8$, $A_4$ with and without anomaly, twin algebras and twin phases, and the Clifford-hierarchy gates in quantum doubles of dihedral groups. All arithmetic is exact and AlgebrasTQD can be used with any finite group $G$ and 3-cocycle $\omega$, limited only by the time and size of the calculation. The code is available at this https URL

We study whether nearby homogeneous cosmological histories approach or move away from the Genesis and contracting solutions. We also determine how an uncertainty in the scalar velocity at a later stage affects the inferred initial velocity. In the Genesis models, including the gravitational dynamics can substantially change this relation. We find that convergence of homogeneous solutions can occur even around backgrounds with unstable perturbations, and therefore attraction of the background is not sufficient to establish that a cosmological solution is physically viable. In the contracting model, one power-law solution attracts nearby trajectories, whereas another repels them. The strength of these effects depends on how the cosmological state is parametrized. Their evolution through the subsequent bounce is not considered here.

We introduce a semi-agnostic modeling framework for black hole binary inspirals beyond general relativity that fixes the field content without selecting a particular theory. We expand in the post-Newtonian order and in powers of $l/L$, where $l$ is the characteristic length-scale that controls deviations from General Relativity and $L$ is the characteristic length-scale of the system. We focus on the case of a massless scalar field coupled with the metric, at leading post-Newtonian order. Taking into account the scaling with $l/L$ of both the interaction terms and the quantities that characterise the sources, we find that the inspiral can be described in terms of a small number of universal parameters. Such a parametrisation is suitable for obtaining combined bounds from multiple sources and performing searches for specific fields.

Matter-wave interferometers (MWIs) are a genuine quantum probe of dark matter (DM), being sensitive to DM-induced phase shifts and loss of contrast. Treating the interferometer as an open quantum system and the dark sector as its environment, we derive an open EFT to study these signatures using the Schwinger-Keldysh formalism. Starting from a microscopic theory, we derive the interferometer's effective action, which we then project onto the many-body worldline sectors relevant for atom interferometers and MWIs employing mesoscopic objects, i.e., composite-particle interferometers (CPIs). This construction clarifies the heavy-probe limit and organizes phase shifts and decoherence as distinct real-time components of the influence action: the retarded kernel controls the causal response of the environment, while the Keldysh kernel controls environmental fluctuations and hence loss of contrast. At leading order, these kernels reduce to: (i) DM bubble diagrams in the case of four-point interactions, and (ii) dressed in-medium mediator propagators in the case of long-range interactions, where the dressing is provided by the DM environment. Within this framework, we study the signatures of elastic spin-independent sub-GeV DM scattering off nucleons and find that: (i) owing to Bose enhancements below the DM wave-particle boundary, ambitious CPIs could probe Higgs-portal DM with mass below 1 eV in an unexplored region of model space; (ii) for DM coupled to nucleons via a light hadrophilic scalar, the DM background can drive the mediator's in-medium mass tachyonic over much of parameter space; outside these regions, CPIs could probe provide powerful probes of light DM.

Both Weakly-Interacting Massive Particle (WIMP) and Axion-Like Particle (ALP) are attractive particle dark matter candidates. In this paper we consider a scenario in which both WIMP and ALP are present. The two species interact via a non-derivative quadratic coupling modeled on the axion-nucleon interaction in QCD. We show that even a small (Planck-suppressed) coupling can drastically modify the cosmological history of both species. In the very early Universe, the ALP field acquires a vacuum expectation value, and the system undergoes an inverse phase transition at a temperature around or below the weak scale. WIMPs may remain in equilibrium far beyond the usual freeze-out time, eventually decoupling via a novel "coherent freeze-out" mechanism. WIMP annihilation cross sections several orders of magnitude above the canonical thermal target value can be consistent with the observed relic density in the coherent freeze-out scenario. At the same time, coherent oscillations of the ALP field do not begin until after the phase transition, and the initial conditions for these oscillations are set by the thermal physics of the phase transition. The ALP relic density is then predicted unambiguously in terms of the WIMP mass and coupling. We term this scenario "ALP miracle", since the observed relic density is obtained when these parameters are around their natural values, the weak and the Planck scale. The ALP miracle scenario prefers the ALP mass above about $10^{-3}\,\mathrm{eV}$, far larger than other popular ALP dark matter models. The ALP miracle parameter space overlaps with a currently allowed region of the QCD axion band, a large fraction of which is expected to be probed by the proposed IAXO experiment.

We develop a projection formalism for four-dimensional massless scattering amplitudes based on the construction of harmonics on multi-particle phase space. This provides a generalization of the partial-wave decomposition valid for any number of particles. Applying this formalism to calculations of on-shell cuts, we show that any cut calculation can be rewritten as a sum of products of projected coefficients and derive a condition for this sum to vanish. Using the on-shell renormalization group equation in combination with our projection formalism, we derive new non-renormalization theorems for effective field theories valid at general loop orders.

We investigate the interplay between confinement and Krylov complexity in the long-range transverse-field Ising chain, where power-law interactions $\sim 1/r^{\alpha}$ generate an effective confining potential between domain walls without an explicit symmetry-breaking field. Using the time-averaged Krylov state complexity after a quench as a diagnostic, we find that the complexity is strongly suppressed and only weakly system size-dependent at small $\alpha$ and small transverse field $h_x$, where confinement is strongest, and crosses over to rapid growth at larger $\alpha$, where it increases with system size. Within the confined regime, the oscillation frequencies of the complexity agree with the meson masses of an effective two-kink model, with the dominant frequency growing linearly with the lightest meson mass and the time-averaged complexity decreasing approximately as its inverse square. We further study operator Krylov complexity at finite temperature using the Wightman inner product. At low temperature, it exhibits bounded oscillations that directly encode the meson spectrum, whereas this structure is progressively washed out as the temperature increases. These results identify Krylov complexity as a spectroscopic probe of interaction-induced confinement and reproduce, in a microscopic lattice model, the hallmarks of Krylov complexity identified in confining field theories.

We show that, for most gauged scalar fields, the scalar potential can be systematically organised into an ordered tree of increasingly symmetric operators. The intermediary global symmetries represented in the tree allow at tree-level a subset of the scalar field components, the custodial scalars, to be technically lighter than the scalar mass scale, or even exactly massless (so-called accidents). We further demonstrate, using the notion of moment maps, that such structure can be related to the Higgs branch of supersymmetric theories, allowing the use of powerful symplectic geometry tools in studying the vacuum of non-supersymmetric scalar potentials. We present various applications, from an example of a non-abelian accident to a Standard Model-like Higgs sector generated from custodial scalars.

Grokking is a delayed transition from memorization to generalization in deep neural networks (NNs). We examine whether scaling and renormalization group (RG) theories can describe this dynamics, establishing a quantitative physical framework for neural generalization transitions. For modular addition on a two-layer NN, an ensemble of trained models defines an order parameter (the squared weight norm) and fluctuation proxies for the heat capacity and the susceptibility, with training epoch and weight decay serving as thermal and field control coordinates. To bridge neural architectures to real-space RG, we construct an exact coupling matrix via a second-order expansion of the loss with respect to per-neuron norms and use its Fiedler vector to define an effective geometry. This spectral ordering yields a monotonic spatial decay of coupling strength that is absent under random permutations. We then execute real-space RG by block-spin aggregating adjacent neurons and parameterizing the coupling matrix into off-diagonal band strengths. Linearizing this local RG map yields relevant scaling exponents and five derived critical exponents, with direct empirical estimates matching RG predictions within 1-10%. Overall, our framework provides a consistent effective description of grokking, suggesting the scaling and renormalization group formalism as a tool to investigate NN dynamics.

We prove that mutation-acyclicity of quivers on four vertices is decidable. If a four-vertex quiver $Q$ is mutation-acyclic, then some sequence of mutations takes it to an acyclic quiver without any arrow multiplicity exceeding $\operatorname{max}\{\operatorname{ht}(Q),m(Q)\}$, where $\operatorname{ht}(Q)$ is the largest arrow multiplicity of $Q$ and $m(Q)$ is the Markov invariant of Fomin and Neville evaluated on the totally proper wiggle class of $Q$. A search within a computable form of this bound decides mutation-acyclicity of four-vertex quivers with $O(h^{24})$ states for input height $h\ge1$. Together with a theorem of Caldero and Keller, it also decides mutation equivalence of two four-vertex quivers up to relabeling whenever one of them is mutation-acyclic. We conjecture that $\operatorname{ht}(Q)$ alone suffices as the bound, which would reduce the search to $O(h^{6})$ states, and we verify this for all four-vertex quivers of height at most three. In the other direction, for every $n\ge4$ we construct an affine family of connected $n$-vertex quivers whose nonnegative even members are mutation-acyclic and whose positive odd members are mutation-cyclic. Consequently, unlike in rank three, no fixed finite Boolean combination of polynomial equations and strict polynomial inequalities in the arrow weights (i.e., a semialgebraic criterion) decides mutation-acyclicity of $n$-vertex quivers, even allowing polynomials that are not mutation-invariant. Finally, we correct the criterion for four-vertex quivers of height at most two stated in our earlier paper, and recover its classification counts with short certificates. The calculations in the proof of the height bound are verified by computer for all values of their parameters.

We investigate inflationary magnetogenesis driven by a non conformal coupling in the case of large-field $\alpha$-attractor inspired models. We introduce potential-dependent electromagnetic coupling functions $f(\phi)$ that reproduce the scaling $f\propto a^p$ during slow roll. Starting from Starobinsky inflation, we analytically derive explicit couplings for $(E)$-model and $(T)$-model potentials. We then study how these functions affect electromagnetic amplification in the regime of non-minimal coupling between the inflaton field and the Ricci scalar $\phi^2 R$. In particular, we compare different values for the coupling strength $\xi$, accounting for electromagnetic backreaction and Schwinger pair production through the gradient expansion formalism. Our numerical results show that the non-minimal coupling affects the evolution of the electromagnetic sector and the resulting magnetic-field amplitude. We find that, within the analyzed configurations, minimally coupled and $\xi>0$ scenarios are compatible with the adopted CMB constraints, yielding conditional present-day magnetic-field estimates of order $10^{-23}$ G.

Spin-orbit coupling makes the mean position of a spinning particle transform anomalously under a Lorentz boost, with an additional shift known as a side jump. In this work, we develop a trace method to derive an exact but compact transformation kernel for the probability center of a general three-dimensional Dirac wave packet under an arbitrary finite Lorentz boost. The expression includes the physics of finite momentum width, spin-momentum correlations, and coherent phase structure without a semi-classical expansion. We identify texture-dependent side jumps even at fixed initial mean momentum, spin, and orbital angular momentum, as well as a non-monotonic mass dependence for Gaussian packets at fixed momentum width. The explicit kernels further resolve the spin and orbital angular-momentum contributions for massive and massless fermions, providing a kinematic basis for covariant particle-based spin transport that retains wave-packet information.

What is the intrinsic reason of emergence of macroscopic order parameter out of microscopic systems? In this article, we apply to physics the collective predictive coding (CPC), the theory of language/symbol emergence. As our language emerges out of our human communications, physical macroscopic order parameter emerges out of communications among atoms --- each atom is regarded as an agent who predicts each other's behavior through inference. We present three examples of classical spin systems: two-dimensional ferromagnetic and antiferromagnetic Ising models, a two-dimensional spin model with competing stripe orders, and the fully connected Sherrington--Kirkpatrick spin-glass model. In all of these examples, proper macroscopic order parameters including the replica overlap emerge automatically through communication informatics of CPC, only from observed spin configurations. We call the learned macroscopic coordinate the collective predictive order. Furthermore, for the ferromagnetic and antiferromagnetic Ising models, the equilibrium distribution of the collective predictive order yields an emergent potential with the symmetry and thermal evolution characteristic of a Landau potential. Notably, the emergence of the collective predictive order does not refer to any structure of microscopic Hamiltonians/Lagrangians. Thus our ``Atom-as-an-Agent" (AaaA) principle offers a completely data-driven method for the emergence of macroscopic order, that is, machine learning physics.

We formulate a description of self-gravitating Schrödinger--Poisson (SP) solitons in curved spacetime using a local inertial frame centered on the soliton. In the weak-field and non-relativistic regime, the effect of the background spacetime is incorporated into the SP equations through an effective tidal potential determined by the spacetime curvature. We apply this framework to a soliton moving along a circular equatorial geodesic around a Kerr black hole. The tidal field contains stationary and time-dependent quadrupolar components, which induce a deformation of the soliton and an associated mass quadrupole moment. We formulate the quadrupolar response in terms of the tidal deformability and the dimensionless Love number. We then estimate the gravitational radiation associated with the rotating deformed soliton and evaluate the corresponding characteristic strain for representative soliton sizes and orbital radii. A comparison with representative sensitivity curves of planned space-based gravitational-wave detectors identifies parameter regimes in which the estimated characteristic strain may lie within their sensitivity ranges. Our framework provides a systematic basis for investigating the tidal response of self-gravitating bosonic solitons in black-hole spacetimes and their potential gravitational-wave signatures.

We study Frobenius action on the trigonometric Ruijsenaars--Schneider system associated with cotangent bundles to complete flag varieties. The construction continues the program initiated by Etingof and Varchenko and developed in quantum $K$-theory by the author and Smirnov. We explain how the monodromy eigenvectors recently constructed by Tarasov and Varchenko fit the quantum/classical duality between Bethe equations and tRS level sets. At a root of unity, all partial shifts cancel in the Newton--Adams combinations of the tRS Hamiltonians, leaving precisely the operators obtained by Frobenius action. This proves compatibility of the joint eigenvalue problem with Frobenius action in every rank and at every root order.

One of the central expectations of quantum gravity is the resolution of the curvature singularities predicted by General Relativity, potentially giving rise to effective black-hole geometries that are regular inside the horizon. We show that, for a very broad class of regular black holes including several known solutions, the quasinormal-mode spectrum in the large-overtone limit differs discontinuously from that of a Schwarzschild black hole, even when the regularization parameter is arbitrarily small. The real part of the quasinormal frequencies vanishes asymptotically, rather than approaching a nonzero constant, as generically occurs in General Relativity and other theories predicting black holes with curvature singularities. The asymptotic black-hole spectrum thus retains a distinctive imprint of singularity resolution. We establish these results analytically through a systematic formulation of the monodromy method and confirm them numerically. We interpret this phenomenon as a manifestation of black-hole spectral instability, whereby arbitrarily small deviations from vacuum General Relativity produce nonperturbative changes in the large-overtone spectrum.

We construct Lax pairs with a spectral parameter for a family of models of interacting tops associated with the $\rm{BC}_n$ root system. The construction uses the parametric $\rm{BC}_n$ associative Yang--Baxter equations, which involve both quantum $R$-matrices and boundary $K$-matrices satisfying the reflection equation. Special cases yield spin generalizations of the elliptic Calogero--Inozemtsev model. We also show how quantum Hamiltonians of interacting tops, combined with the Polychronakos freezing prescription, reproduce recently introduced anisotropic quantum long-range spin chains of Haldane--Shastry--Inozemtsev type.

We analyze the on-shell content of the local Lorentz covariants of the five-dimensional spin-zero Duffin-Kemmer-Petiau algebra in elastic spin-zero-spin-$1/2$ scattering. The scalar, vector, and tensor structures are reduced explicitly to the two independent amplitudes allowed by the physical parity-conserving process. The two scalar structures are proportional on shell, the genuine nonminimal vector $i[P,\beta^\mu]$ is longitudinal and vanishes after contraction with the conserved fermion vector current, and the projected vectors $P\beta^\mu$ and $\beta^\mu P$ each reduce to one half of the minimal-vector contribution. The tensor covariants introduce no additional on-shell directions. We discuss the consequences for the four invariant choices used in meson-nucleus optical-potential calculations and show that different nuclear equations can arise only after on-shell-equivalent representatives are promoted to optical operators and iterated beyond the order at which the elementary impulse approximation is fixed. These results provide a systematic basis for distinguishing genuinely independent elementary amplitudes from representation-dependent effects generated by the optical embedding.

We study the covariant Wigner function for the spin-$3/2$ Rarita-Schwinger field and its relation to the spin density matrix and the mean spin polarization vector. We obtain a formula relating the covariant Wigner function to the mean spin vector for a weakly interacting field. We then calculate the mean spin vector induced by thermal vorticity, thermal shear, and spin potential at local thermal equilibrium in the Belinfante, canonical and GLW pseudo-gauges. The resulting formula can be used to predict the spin polarization of spin-$3/2$ hadrons, such as the $\Omega^-$, in heavy-ion collisions. Differences in the thermal-shear induced polarization in the canonical and Belinfante pseudo-gauges make it possible to use local spin polarization measurements of the $\Omega^-$ to discriminate between the two pseudo-gauges.

We investigate single-field inflation models that can produce large curvature perturbations at sub-Mpc scales while remaining consistent with cosmological observations across multiple regimes. We highlight the tension between generating enhanced small-scale perturbations and maintaining agreement with the scalar spectral index $n_s$ measured by ACT and SPT, which disfavors several conventional single-field inflation models. To address this issue, we propose three viable classes of inflationary potentials: bump/dip-modified, step-like, and double-inflection-point. Each can sufficiently amplify primordial perturbations while predicting $n_s$ that lie within the $2\sigma$ range of current observational bounds. In addition, we examine the primordial power spectra predicted by these models and constrain their parameter spaces using cosmological observations at different scales, including CMB, BAO, PBH, PTA, and CMB-distortion. Our results show that some of the models considered can simultaneously satisfy the ACT/SPT constraints and account for the current PTA observations, although doing so while respecting the constraints on the PBH abundance remains challenging.

This note establishes a rank reduction for RLL-realization of quantum affine supergroups of orthosymplectic type. In the rational limit, this provides a more conceptual proof of the analogous result for orthosymplectic superYangians from arXiv:2311.18818. We also determine the exact scalar prefactor appearing in the evaluation of the universal $R$-matrix, refining the main result of arXiv:2408.16720.

Rotating black holes offer powerful probes of gravity, but constructing their geometries beyond vacuum general relativity remains challenging. Here we develop a unified framework for constructing stationary, axisymmetric deformations of Kerr from their tidal curvature and physical sources. As a demonstration, we obtain closed-form solutions for six higher-curvature interactions at leading order in the couplings and for generic subextremal spin, without slow-rotation expansions, numerical fitting, or mode decomposition. The solutions pass stringent checks against the field equations, previous mode-by-mode constructions, and known analytic predictions for horizon quantities. We find that the chirality of the interactions organizes the analytic structure of these solutions, a pattern that likely extends to a large class of purely metric higher-curvature theories. Our framework opens a systematic route to black-hole geometries in beyond-Einstein theories and perturbative astrophysical environments. These geometries provide a foundation for strong-field tests with gravitational waves and other observational channels, such as black-hole images and X-ray spectroscopy, as well as theoretical studies of near-extremal black holes.

Compilation of two-qubit Fibonacci anyon gates is complicated by leakage ($\ell$) from the four-dimensional computational sector to the fifth state of the fusion space. We construct a GPU pipeline to find low-leakage representatives of the $[\text{CNOT}]$ class on different subsets of the braid group. We exhaustively explore the full positive five-generator alphabet up to $L=21$ with minimal leakage $\ell\sim10^{-3}$ and error $d_{[\text{CNOT}]}\sim 10^{-8}$ on it. To reach larger lengths, the search space is successively narrowed to three-generator, boundary-$\sigma_3$, and palindromic braids, and by considering only the Garside left normal form. These optimizations allow us to reach $L=72$. For $L=69$, a braid with $\ell\sim10^{-10}$ and $d_{[\text{CNOT}]}\sim 10^{-10}$ is found. In addition to directly enumerating optimal braids, we proved that $\ell\in\mathbb Z[\varphi]$, and any nonzero leakage obeys an exponential lower bound in the braid length, and obtained the limit laws $F_{3,\infty}(\ell)=\ell^2$ and $F_{5,\infty}(\ell)=\ell^4$ for our sets. Using asymptotics for the distribution function $F$, we showed that for sufficiently long braids, the asymptotics of exhaustive search are better than the asymptotics of random sampling for finding low-leakage braids.

We revise and extend the algorithm of F. G. Scholtz and B. Chakraborty [J. Phys. A: Math. Theor. 46 (2013) 085204] to compute the finite Connes' distance between normal states on the Moyal plane and fuzzy sphere, using the Hilbert-Schmidt operatorial formulation. The original expression gives only a lower bound. The revised expression involves an infimum over the "transverse" component $\Delta\rho_\perp$ of the algebra element in addition to the "longitudinal" component $\Delta\rho$, identified with the difference of density matrices representing the states. Determining the transverse component for which the infimum is reached remains non-trivial, but under rather generic conditions the distance is proportional to the Hilbert-Schmidt norm of $\Delta\rho$. We also determine an upper bound by adapting the approach of P. Martinetti and L. Tomassini [Commun. Math. Phys. 323 (2013) 107-141], and look for an optimal element saturating it. For the Moyal plane, such an element is obtained through the limit of a sequence of finite-dimensional projections of the representative of a multiplier-algebra element onto subspaces spanned by Dirac eigen-spinors. In contrast, on the fuzzy sphere the upper bound, given by the geodesic distance on a commutative sphere, is never reached for any finite $n$-representation of $SU(2)$. For maximal non-commutativity ($n=1/2$), the finite distance coincides with the lower bound and the transverse component plays no role. This changes from $n=1$ onwards, where the revised algorithm improves the estimate and gives an almost exact result. The contrasting features of these spaces become transparent through the analysis in the eigen-spinor bases of their Dirac operators.

We study modular symmetries in non-supersymmetric heterotic string theories on toroidal backgrounds with Wilson line modulus, constructed by stringy Scherk-Schwartz compactification. In particular, we focus on a subgroup of the T-duality group $O(D+16,D,\mathbb{Z})$ with $D=2$ given by an outer automorphism of the Narain lattice, which can be mapped to the Siegel modular group $\mathrm{Sp}(4,\mathbb{Z})$. We classify the modular symmetries and a $CP$-like symmetry on $T^2$ and its orbifolds with symmetric and asymmetric orbifold twists. It turns out that the non-supersymmetric heterotic string theories only enjoy a part of modular symmetries in contrast to supersymmetric ones. Furthermore, the gauge symmetry is maximally enhanced at fixed points of modular symmetries on $T^2$ on which we analyze the vacuum structure of eight-dimensional tachyon-free vacua.

In this paper, we relate the MacDonald index of a 4d $\mathcal{N}=2$ SCFT with the Hilbert series of the arc space of the Zhu's $C_2$-algebra of the corresponding Schur VOA. Using this, we conjecture a simple formula for the MacDonald index of $(A_1,D_{2n+1})$ Argyres-Douglas theory. We perform checks of the formula against the known Schur limits and RG flows. To match the Schur limit, we prove new $q$-series identities.

$T\bar{T}$-deformed CFTs are known to possess nonlocal Virasoro$\times$Virasoro symmetries, whose currents depend on integration of the stress tensor. Unfortunately, these symmetries do not act in a tractable manner on the undeformed operators, which are defined by evolving local operators in the undeformed CFT on a constant-time slice using the deformed Hamiltonian. In this paper, we explicitly construct two distinct classes of operators: (i) dressed operators, which are primary with respect to the Virasoro$\times$Virasoro symmetries, and (ii) physical operators, a new type of operator that depends on fundamental fields at specific points and is therefore local at the classical level. While dressed operators preserve the conformal symmetry structure, they are themselves nonlocal in the sense that they depend on integrated quantities. Physical operators, by contrast, depend only on local fields. Moreover, physical operators can be expressed in terms of dressed operators, and this relation can be promoted to a definition of such operators at the quantum level. This enables the use of Virasoro$\times$Virasoro symmetries to systematically compute physical quantities such as correlation functions. We calculate the two-point correlation functions of these operators in momentum space and find that our results align with both string theory predictions and field theory calculations. Additionally, we explore the relationship between physical operators and alternative operator definitions proposed in the literature.

We compute the gluonic contribution to the gravitational form factors of the proton using the improved holographic QCD model, in which the proton is described in terms of bulk Dirac fermions. Model parameters are constrained using lattice and phenomenological input, allowing us to obtain estimates for the gravitational form factors and to compare them with results from other approaches. Using the extracted form factors, we analyze mechanical properties of the proton, including pressure and shear distributions. We obtain estimates for the mechanical and mass radii of the proton, $\rho_{\text{mech}} \approx 0.63$ fm, $\rho_{\text{mass}} \approx 0.65$ fm, and compare these values with radius estimates from other nonperturbative studies.

We present the interpolating conformal algebra between the instant form dynamics (IFD) and the light-front dynamics (LFD) in $(1+1)$ dimensions, along with a $4\times4$ interpolating projective spacetime matrix representation. While there are six generators in the $(1+1)$ dimensional conformal algebra, the number of kinematic and dynamic generators dramatically changes in LFD, maximizing (minimizing) the number of kinematic (dynamic) generators to four (two) with respect to two (four) kinematic (dynamic) generators in IFD, as well as in any other forms of dynamics between IFD and LFD. It confirms and signifies the utility of LFD, saving substantial dynamical efforts in solving the $(1+1)$ dimensional quantum field theories. We also present $2\times2$ Pauli matrix representation of $(1+0)$ and $(0+1)$ conformal groups, and creation/annihilation operators of quantum simple harmonic oscillator representations of $(1+0)$ dimensional conformal groups.

We advocate for a holographic definition of thermodynamic pressure and volume for black holes based on quasi-local gravitational thermodynamics. When a black hole is enclosed by a finite timelike boundary, York's quasi-local first law includes a surface pressure conjugate to the boundary area. Assuming the existence of a holographically dual theory living on this boundary, these geometric quantities correspond to the pressure and volume of the dual thermal system. In this work we focus on static, spherically symmetric black holes, for which these quantities reduce to global thermodynamic variables. The holographic volume provides a notion of system size, allowing extensivity to be defined in standard thermodynamic terms, and it yields a definition of the large-system limit. For the asymptotically flat case, we show that, in the canonical thermodynamic representation, small Schwarzschild black holes are non-extensive, whereas large black holes become extensive in the large-system limit. A similar conclusion applies to Anti-de-Sitter Schwarzschild black holes, with the difference that the quasi-local energy of the large black hole also becomes extensive in the large-system limit. Before this limit, the energy decomposes into subextensive and extensive contributions, and we derive an explicit expression for the extensive part as a function of the finite volume and entropy.

We construct an $\mathcal N=2$ supersymmetric Grassmannian representation of tree-level wavefunction coefficients (WFCs) by combining Grassmannian representations of energy discontinuities with an inversion formula. Since the orthogonal Grassmannian captures homogeneous solutions of the spinor conformal Ward identities, while current WFCs satisfy inhomogeneous Ward identities, the full WFC is obtained by reconstructing the energy-dependent prefactors from a basis of discontinuities. We first demonstrate this mechanism at three points, where the triple discontinuity determines the transverse current WFC and admits a supersymmetric uplift. At four points, we invert a spanning set of five current discontinuities and embed the result in momentum superspace using super-orthogonal-Grassmannian invariants generated by $\hat\delta(C\Omega\Xi^I)$. This yields the full four-point super WFC in Grassmannian form. We show that the two orthogonal-Grassmannian branches organize distinct supersymmetric invariants and reduce, in the flat-space limit, to different helicity superamplitudes.

The so called $\kappa$-Minkowski spacetime is a prominent candidate for a quantum gravity space, and its properties have been the subject of investigation over the last thirty years. In this work, we propose classical electrodynamics equations that satisfy the dispersion relation derived from the bicrossproduct approach to the $\kappa$-Poincaré algebra. We explore the conservation laws, a new gauge symmetry, and the solutions for the scalar and vector potentials under the influence of the deformed parameter. The characteristics of the vacuum as a material medium are shown in terms of the refractive index and the group and phase velocities. The Green's functions associated with the wave operator in the bicrossproduct basis are calculated in both the retarded and advanced prescriptions. Finally, we discuss how the deformation scale affects the causality of the theory.

Following Veltman's approach, we formulate and discuss a general cutting equation for QED in a plane-wave background. We apply the corresponding cutting rules to justify the connection between the two-loop radiative corrections to elastic electron scattering and the rate of the trident process in a constant crossed field. Among other terms generated by cuttings, some field-induced radiation corrections also emerge, which contain UV-finite contributions from loop insertions into external legs. As a byproduct, we compare the previously published results for the trident process in a constant crossed field and present a complete analytical expression for direct and exchange contributions to its rate, which is resolved in the spin of the initial electron. Our findings establish that although total rates can be reliably extracted from higher-loop by applying the cutting rules, reconstruction of differential rates requires additional care. The cutting rules apply to any loop order and may be extended to nonperturbative regimes.

A novel inequality involving the residual information and genuine multi-entropy was proposed in \cite{Balasubramanian:2025hxg} for tripartite holographic pure states, using which it was argued, that purely GHZ-like tripartite entanglement is not allowed in holography. In this work, we generalize this holographic signal inequality to mixed states. In a minimal extension, we compute the reflected genuine multi-entropy following \cite{Yuan:2024yfg} and find a class of holographic geometries that violate this minimally extended inequality due to a vanishing Markov gap. We can symmetrize this prescription, where instead of computing the residual information on the given mixed state $\rho_{ABC}$, we compute it on its canonical purification. The inequality is restored on the canonically purified state. Finally, we conjecture a new inequality for tripartite holographic states and give supporting evidence.

We elucidate the physical nature of ghosts above the multi-particle threshold by contrasting them with unstable particles in quantum field theory. We first consider the asymptotic formulation, where ordinary positive-norm one-particle states can be unstable and decay, whereas ghosts survive asymptotically without decaying, yet admit no particle interpretation due to interference with the multi-particle component which masks the negative-norm one-particle state. This distinction originates from two different analytic structures of the dressed propagator, whose complex conjugate poles lie on the first or second Riemann sheet in the ghost or ordinary case, respectively. Ghost resonances are, in principle, phenomenologically distinguishable from ordinary ones, being narrower and exhibiting weaker interference between positive- and negative-energy peaks. We then formulate the quantum field theory in a finite interval of time and, working within a suitable approximation for the dressed propagator, find that finite-time effects amplify differences in the resonant behavior and give rise to new features, such as higher peaks in ghost resonances. Distinct temporal regimes are also identified: for times shorter than the inverse width, an approximate free-particle description is valid; at later times interactions and interference dominate, leading to decay or multi-particle masking. This study supports the absence of freely propagating ghost particles in the asymptotic limit.

We reconsider the "fortuity'' mechanism in the D1D5 CFT focusing on the K3 symmetric orbifold. Going beyond the counting of BPS states, we investigate perturbatively how the explicit form of the BPS cohomologies is modified by the twist-two deformations. We calculate the action of the supercharges in the sector $(h,j)=(1,0)$ for different values of the central charge and derive explicit expressions for the primary states. Equipped with this information, we compare some protected three-point couplings in the free and the gravity regime. We show that agreement between the two descriptions imposes non-trivial constraints on the identification of monotone and fortuitous states. In particular, we argue that the map relating theories with different values of the central charge must and can be defined so as to commute with the supercharges that define the cochain complex. We then study the three-point correlators between the fortuitous and monotone states identified in our analysis to assess whether the two sectors are dynamically decoupled. We find an explicit example of a non-vanishing coupling between two monotone and a fortuitous state, providing evidence that the two sectors are dynamically connected.

Multi-winding Wilson loops (Wilson spools) in 2+1-dimensional gravity can reproduce one-loop partition functions of local free fields in the bulk. Bulk free fields have a second-quantized Fock space, and the AdS/CFT correspondence suggests that the associated multi-particle sectors are dual to multi-trace primaries in the CFT. In this note, we use standard symmetric-function methods to show that the Wilson spool in thermal AdS$_3$ can be recast as a sum over $single$-winding Wilson loops -- one for each multi-trace primary. In a companion paper to appear, we will view Wilson networks and TQFTs as the natural language of non-perturbative bulk quantum gravity. The present note illustrates how this can apply to $local$ bulk fields, and not just defects: a bulk (generalized free) field is to be viewed as a full tower of multi-trace Wilson lines. We further show that the $SL(2)$ descendants of each multi-trace primary, together with the boundary gravitons of the AdS$_3$ background, correctly reproduce the full Virasoro character of each module. In this language, the role of a light insertion on a heavy primary is played by a topological Verlinde line. This allows us to obtain a "microscopic" description of one-loop determinants on the smooth BTZ handlebody. A spatially wound probe Wilson line on a torus with a contractible $thermal$ cycle can be traded for a Verlinde line inserted on a dense family of heavy Polyakov loops -- with the roles of the two cycles exchanged, so that the $spatial$ cycle is now contractible. The vacuum row of the modular $S$-kernel acts as the (approximate) density of the heavy primaries. This reinforces the case made in arXiv:2601.18775 that a smooth horizon is a stand-in for an ensemble of quantum states, each produced by a heavy Wilson line that appears as a singular horizon in the semi-classical limit.

The canonical and path integral quantization for non-linear electrodynamics in the presence of an external electromagnetic (EM) field are proposed in this work. The EM background field is introduced expanding a general lagrangian of a generic non-linear electrodynamics around the background for small fluctuations of the propagating fields, where we consider up to the quadratic terms in the propagating fields. As consequence, we obtain an electrodynamics linearized by the presence of the external EM field. Thereby, we study the canonical quantization calculating the energy for the ground state of the linearized EM field in terms of an external magnetic field. The microcausality of the model also is discussed through the Pauli-Jordan function. We also define a generating functional for the linearized EM field, in which, in the Coulomb gauge, the Green function of the model is obtained, and we can construct the perturbative formalism like in standard quantum field theory (QFT). As application of the perturbation theory, the effective potential at one loop is calculated for the linearized EM field coupled to a complex scalar field. We apply the results in the case of the Modified Maxwell ED.

We present a systematic study of finite, non-split 2-group symmetries with a non-trivial Postnikov class, focusing on the simplest example with a $\mathbb{Z}_2$ 0-form symmetry and a $\mathbb{Z}_2$ 1-form symmetry, intertwined together by the non-trivial Postnikov class in $H^3(B\mathbb{Z}_2;\mathbb{Z}_2)\cong\mathbb{Z}_2$, denoted by $\mathcal{G}$. We classify the anomalies of this 2-group symmetry for physical theories in $d$-dimensional spacetime, by computing the oriented bordism groups $\Omega_{d+1}^{\rm SO}(B\mathcal{G})$ and the spin bordism groups $\Omega_{d+1}^{\rm Spin}(B\mathcal{G})$ for $d\leq 5$. For the case of $d=3$, we investigate the Symmetry TFT/TO of the 2-group symmetry $\mathcal{G}$ using the language of fusion 2-categories, as well as (3+1)D TQFT actions, for the cases without or with the 2-group anomaly classified by $\operatorname{Hom}\left(\widetilde{\Omega}_4^{\rm SO}(B\mathcal{G}),U(1)\right) \cong H^4(B\mathcal{G};U(1))\cong\mathbb{Z}_2$. We classify the minimal topological and physical boundary conditions of the Symmetry TFTs, and carry out the categorical Landau paradigm for such a non-split finite 2-group.

As was shown in arXiv:2609.14497, the 2-loop ${\rm T}^{-1}_2$ correction to the partition function of M2 brane wrapping $\mathrm{AdS}_2\times S^1$ inside $\mathrm{AdS}_4\times S^7/\mathbb{Z}_k$ vanishes. This is in agreement with the localization prediction for $\frac{1}{2}$-BPS circular Wilson loop in ABJM theory interpreted in the grand-canonical ensemble according to the conjecture of arXiv:2505.21633. Here we study the ${\rm T}^{-1}_2$ correction for the M2 brane instanton wrapping $S^3/\mathbb Z_k\subset S^7/\mathbb Z_k$ dual to the leading non-perturbative large $N$ contribution to the ABJM free energy on 3-sphere. The 1-loop instanton M2 brane partition function was shown in arXiv:2307.14112 to match the localization prediction. The 2-loop computation is complicated by the presence of bosonic and fermionic zero modes, which requires to integrate over the collective coordinates. The collective-coordinate Jacobian provides a non-trivial ${\rm T}^{-1}_2$ contribution in addition to the one of the quartic interaction vertex in the M2 brane action. We find that for $k>2$ the the quartic vertex contribution is cancelled by the Jacobian contribution for the choice of the separation of the fermionic zero modes from the quantum fields under which these form separate supermultiplets. Assuming the supersymmetric prescription for the paired zero modes we show that the cancellation persists after integration over the collective coordinates. Thus the ${\rm T}^{-1}_2$ correction vanishes, in agreement with the grand-canonical ensemble prediction. For $k=1$ the quartic vertex contribution reduces to equation-of-motion terms as in the case of ${\rm AdS}_3 \subset { \rm AdS}_7 \times S^4$ in arXiv:2511.22306 which is related to $S^3 \subset {\rm AdS}_4 \times S^7$ case by a formal analytic continuation.

We study the asymptotic behavior of large-spin operators in conformal field theories by placing the theory on a pp-wave background. In this setting, the diagonal three-point coefficients involving two large-spin operators and a light operator are mapped to thermal one-point functions on the pp-wave. We predict these one-point functions in the two complementary regimes of high and low temperature, corresponding respectively to the total twist parametrically above and below an appropriate spin-dependent scale. In particular, at high temperature, we derive their asymptotic behavior using thermal effective field theory, while at low temperature, we obtain universal formulas that can be matched directly to predictions from the analytic bootstrap. We test these results in free theories on the pp-wave. Finally, we solve the large-$N$ $\mathrm{O}(N)$ model numerically, finding agreement with our predictions for both the one-point function and the free energy density in the high- and low-temperature regimes.

We consider the extension of the $\delta$-shift connecting Tr$(\phi^3)$ and pion amplitudes to observables in cosmology. We start in flat-space and explain how one can extract the pion wavefunction and correlators from those of Tr$(\phi^3)$ theory, via a simple kinematic deformation. As opposed to the amplitude, the $\delta$-shift for the wavefunction is no longer a linear deformation on the kinematics since the wavefunction depends on magnitudes rather than the squares of momenta. We "linearize" this deformation by using the frequency representation - allowing us to recast the contribution from any graph as a sum over residues. We then show how this construction extends to de Sitter, and find that, remarkably, in the frequency representation, the pion de Sitter wavefunction and correlators can be formulated in precisely the same way as in flat-space. Amongst other things, this unified formulation reveals a simple universal soft-limit for flat-space and de Sitter, where, in the soft-limit, the wavefunction turns into a sum of wavefunctions in a mixed theory of pions and scalars.

We take a first step in identifying the superconformal quantum mechanics dual to supersymmetric AdS$_2\times$S$^2$ solutions. We focus on the Type IIB solutions of the form AdS$_2\times$S$^2\times$CY$_2\times\Sigma$ that preserve eight supercharges, with $\Sigma$ a two-dimensional space parametrised by $(x_1,x_2)$, and read off their brane content, a D1-D5-NS5 intersection with D3 and D7 flavour branes. We propose that their dual $\mathcal{N}=4$ quantum mechanics is organised over a two-dimensional lattice of cells in $\Sigma$. Namely, it consists of copies of the linear quiver along $x_1$ dual to the U(1)-invariant solutions of the class, one for each interval along $x_2$, with ranks given by the D1 and D5 charges of the corresponding cell. Its central charge agrees with the holographic one at leading order. We show that the family of SL$(2,\mathbb{R})$ non-Abelian T-duals of the AdS$_3\times$S$^2\times$CY$_2$ solutions of massive Type IIA supergravity, constructed in previous work, belongs to this class, as does the SL$(2,\mathbb{R})\times$SU$(2)$ non-Abelian T-dual of AdS$_3\times$S$^3\times$CY$_2$, and work both out in detail. For both we propose global completions that introduce additional D3 and D7 branes. Along $x_2$ we interpret the fundamental strings stretched between the colour branes and the additional D3 and D7 branes as baryon vertices. Finally, we extend this proposal from the non-Abelian T-duals to the solutions of the class whose defining functions factorise in $x_1$ and $x_2$, including those in which the AdS$_2$ and S$^2$ factors have the same warp factor.

We study unifying relations between Yang-Mills theory and the nonlinear sigma model. We extend tree-level amplitude relations expressed through differential operators to off-shell multiparticle currents and obtain one-loop integrand relations by sewing two off-shell legs. This off-shell approach may also apply to higher-loop integrands.

We study planar $\mathcal{N}=4$ super Yang-Mills theory (SYM) on the real projective space $\mathbb{RP}^4$ using integrability, holography and supersymmetric localization, and discuss its relation to orientifolds in string theory. We show that the integrability of $\mathcal{N}=4$ SYM is preserved when combining the $\mathbb{Z}_2$ involution defining $\mathbb{RP}^4$ with the action of the charge conjugation of the SU$(N)$ gauge group. We demonstrate this by calculating one-point functions of non-BPS single-trace operators in various subsectors at weak coupling, which can be mapped to overlaps between energy eigenstates and crosscap states in integrable spin chains. The results exhibit a parity selection rule, a hallmark of integrability, and are given in terms of a ratio of determinants. Building on these results, we conjecture a closed formula for general single-trace operators. We further provide evidence that this construction is holographically dual to a probe orientifold O1-plane in $\text{AdS}_5\times \text{S}^5$ by constructing the corresponding worldsheet crosscap states in the pp-wave limit. Finally, we use supersymmetric localization to compute one-point functions of half-BPS operators exactly at finite $g_{\rm YM}$ and $N$. The computation reduces to two-dimensional Yang-Mills theory on $\mathbb{RP}^2$ twisted by charge conjugation and leads to a matrix integral of a Wishart-Laguerre type. In the planar limit, the exact result interpolates between the weak-coupling gauge-theory result, the strong-coupling supergravity prediction for the orientifold, and the worldsheet crosscap in the pp-wave limit.

We give a fourth-order Calabi-Yau differential operator, which seems to have been missed by earlier tabulations. We draw attention to this operator because of its clear geometric interpretation and evidence that it possesses a rank-two attractor, conjecturally giving a new example of a Calabi-Yau threefold with modularity properties analogous to those of an elliptic curve over $\mathbb{Q}$.

The side jump in the anomalous Lorentz transformation, arising from the spin-orbit interactions, plays important roles in various intriguing physics, such as chiral vortical effects and spin polarization. In this work, the side jump of the spin-half particles is visualized and evaluated ab initio within Dirac theory, through a comprehensive study of the density $\psi^\dagger\psi$ and position $\int d^3x\psi^\dagger \mathbf{x}\psi$ of a wave packet under Lorentz transformation. A compact gamma-matrix representation of the Lorentz boost matrix of a Dirac spinor, $S(\boldsymbol{\eta})= h_{\boldsymbol\eta}^{\mu}\gamma_\mu \check{u}^\nu \gamma_\nu$, has been found. Based on this expression, a trace method has been developed to derive the anomalous Lorentz transformation of the center of the probability density of a spin-half fermion wave packet, and a compact analytical expression for the side jump has been obtained through a transparent and tractable derivation. The momentum width plays an important role in the side jump. For example, for Gaussian wave packets, increasing the width suppresses the displacement, while the side jump at fixed width depends nonmonotonically on mass, reaching a maximum at a finite mass. In the zero-width limit, the side jump takes a compact form perpendicular to both the Pauli-Lubanski vector and the boost direction, recovering the known result for a chiral particle.

The Emergent Universe scenario is based on the assumption that the universe originates from a past-eternal Einstein static (ES) state, subsequently evolving toward an inflationary phase and a hot Big Bang era. Such models are appealing as they provide nonsingular and geodesically complete cosmological histories. However, it has been argued by Mithani and Vilenkin that, even when the ES state is classically stable, certain models can admit semiclassical tunneling channels leading to quantum decay toward configurations of vanishing scale factor. In this work, we investigate the classical and semiclassical stability of the ES regime in the context of Jordan-Brans-Dicke (JBD) theory. We analyze the structure of the Wheeler-DeWitt potential in minisuperspace and study representative semiclassical tunneling channels compatible with the Hamiltonian constraint. We show that, for suitable choices of the JBD potential and model parameters, the ES configuration can be robust against both classical perturbations and the specific semiclassical tunneling channels analyzed in this work. Our results indicate that the quantum instability discussed by Mithani and Vilenkin may be avoided within certain regions of parameter space, while leaving open the possibility of more general tunneling processes beyond the scope of the present analysis.

You may have seen the words "topological recursion" mentioned in papers on matrix models, Hurwitz theory, Gromov-Witten theory, topological string theory, knot theory, topological field theory, JT gravity, cohomological field theory, free probability theory, gauge theories, to name a few. The goal of these lecture notes is certainly not to explain all these applications of the topological recursion framework. Rather, the intention is to provide a down-to-earth (and hopefully accessible) introduction to topological recursion itself, so that when you see these words mentioned, you can understand what it is all about. These lecture notes accompanied a series of lectures at the Les Houches school "Quantum Geometry (Mathematical Methods for Gravity, Gauge Theories and Non-Perturbative Physics)" in Summer 2024.

We construct a unified (quantum) description, by the gauge principle, of gravity and Standard Model (SM), that generalises the Dirac-Born-Infeld action to the SM and Weyl geometry, hereafter called Weyl-Dirac-Born-Infeld action (WDBI). The theory is formulated in $d =4-2\epsilon$ dimensions. The WDBI action is a general gauge theory of SM and Weyl group (of dilatations and Poincaré symmetry), in the Weyl gauge covariant (metric!) formulation of Weyl geometry. The theory is SM and Weyl gauge invariant in $d=4-2\epsilon$ dimensions and there is no Weyl anomaly. The WDBI action has the unique elegant feature, not present in other gauge theories or even in string theory, that it is mathematically well-defined in $d=4-2\epsilon$ dimensions with no need to introduce in the action a UV regulator scale or field. This action actually {\it predicts} that gravity, through (Weyl covariant) space-time curvature $\hat R$, acts as UV regulator of both SM and gravity in $d=4$. A series expansion of the WDBI action (in dimensionless couplings) recovers in the leading order a Weyl gauge invariant version of SM and the Weyl (gauge theory of) quadratic gravity. The SM and Einstein-Hilbert gravity are recovered in the Stueckelberg broken phase of Weyl gauge symmetry, which restores Riemannian geometry below Planck scale. Sub-leading orders are suppressed by powers of (dimensionless) gravitational coupling ($\xi$) of Weyl quadratic gravity.

We investigate stringy constraints on moduli spaces in modular flavor models by analyzing moduli-dependent threshold corrections in heterotic string vacua. While moduli play a crucial role in determining the flavor structure of fermions predicted by modular flavor models, the parameter space in which their vacuum expectation values are allowed has not been fully explored. In this work, within the framework of perturbative heterotic string theory on toroidal orbifolds, we derive constraints on the moduli space and study their systematic behavior. We characterize the stringy constraints in terms of the dilaton, the beta-function coefficients, and the ratio between a complex-structure modulus and a Kähler modulus. It is found that the large value of the modulus controlling the flavor structure, i.e., $\tau\simeq i \infty$, is inconsistent within perturbative string theory, and the self-dual point $\tau=i$ is also disfavored in the large volume regime of toroidal backgrounds. In addition, we discuss the phenomenological implications of these stringy constraints.

We develop a geometric--analytic proof of the reverse isoperimetric inequality for compact AdS black hole horizons in Einstein gravity. At fixed thermodynamic volume, a York slice/Jacobi variation of the gravitational action shows that the spherical Schwarzschild--AdS solution is a strict local entropy maximum against generic non-spherical deformations. The finite global extension is formulated within the hypotheses of the Sherif--Dunsby (SD) rigidity theorem: compactness and spherical topology, the required intrinsic Ricci decomposition $\mathrm{Ric}_h=\alpha\,\mathbf e\!\otimes\!\mathbf e+\beta N$ with $\alpha\neq\beta$, a proper conformal transformation of the hypersurface metric given by $\widetilde h_{ab}=e^{2\varphi}h_{ab}$ that preserves its intrinsic scalar curvature, $\widetilde R\equiv R[\widetilde h]=R[h]\equiv R>0$, and negative sheet expansion. Compactness and $S^3$ topology are built into the class considered here. In the static spherical sector such as Schwarzschild--AdS and RN--AdS, the SD Ricci form is naturally realized while it is an additional restriction for Kerr--AdS case. For the $S^3$ topology, a positive Yamabe type conformal class exists for which the Yamabe theorem provides a representative of positive and constant scalar-curvature. We further restrict to the scalar-curvature preserving conformal transformation required by the SD theorem. The negative sheet expansion is obtained using gravitational focusing. Within this admissible class, a non-round Kerr--AdS horizon cannot maximize the entropy, while fixed thermodynamic volume selects the corresponding round Schwarzschild--AdS comparator. The analysis thus establishes a generic local entropy maximum and a finite global result within the stated geometric rigidity class, identifying the role of gravity in reversing the ordinary isoperimetric problem.

Is the usual treatment of axion dark matter as a classical field reliable? We show that the answer is subtle: the axion field could well be in a quantum state that has no complete classical description, but realistic detectors cannot tell the difference. To see this, we solve a fully quantum model of axion detection using quantum optics techniques. We show that intrinsically quantum effects are washed out by mode averaging or small amounts of noise, and significantly suppressed by the weakness of the axion coupling. Our work exemplifies that there should always be a classical analog for axion dark matter effects, extends to other wave (ultralight) dark-matter candidates, and gives a general method to compute the effects of exotic dark-matter states.

This work aims to present some quantum aspects of a non-commutative (NC) gauge gravity formulation of a Schwarzschild-like black hole constructed via the Moyal twist $\partial_t \wedge \partial_\theta$. Particle creation is estimated for bosonic and fermionic fields using the quantum tunneling method, with divergent integrals treated through the residue prescription. Since the surface gravity is well defined for this configuration, the corresponding emission rates and evaporation lifetimes are also computed. In addition, previously reported results in the literature on gauge gravity Schwarzschild black holes are revisited. Finally, we infer constraints on the NC parameter $\Theta$ from solar system tests.

We analyze the finite-size corrections to the crosscap overlap in the two-dimensional classical Ising model along its self-dual critical line. Using a fermionic formulation, we express the lattice crosscap overlap in terms of Bogoliubov angles and develop a contour-integral approach by analytically continuing the lattice momentum to the complex plane. This leads to a remarkably simple expression for the crosscap overlap, which demonstrates that the finite-size corrections decay exponentially with system size. We further derive an exact analytical formula for the corresponding decay constant and show that it is determined by the complex singularity structure of the Bogoliubov angle.

We derive bounds on the lifetime of an evaporating black hole. The minimum lifetime bound follows from energy conservation and purification, within the framework of `asymptotically semiclassical spacetimes'. The recently derived expression for the Bondi flux of Hawking radiation, together with the expression for the entanglement entropy of Hawking radiation at null infinity, is used to investigate the purification phase after the last semiclassical ray. We discuss the energy-cost of entanglement purification and find a lower bound on the purification time of the black hole, which scales as $M_0^4/\hbar^{3/2}$, where $M_0$ is the initial black hole mass. Additionally, motivated by the Planck luminosity bound in quantum gravity, we include the additional assumption that a Planck mass black hole is metastable. With this assumption, the purification time is in fact exponential in the square of the initial black hole mass, $\tau\sim\sqrt{\hbar}\,\exp(\gamma M_0^2/\hbar)$ with $\gamma>0$, i.e., exponential in its initial area. We find that the redshift exponent is negative in this purification phase, which indicates the existence of a white-hole remnant that releases information slowly. We comment on phenomenological implications for primordial black hole remnants.

We introduce a new series decomposition of the waveform constructed in the spirit of Debye expansions in scattering theory, and we use this to analyse the time-domain response of compact, horizonless bodies to scalar-field perturbations on curved spacetimes. The Debye decomposition separates out direct exterior propagation, surface reflection, and successive transmissions through the interior of a compact body, and it provides an intuitive interpretation of the waveform in terms of geodesic trajectories. By analysing the quasinormal-mode (QNM) content of individual Debye terms, we set out a Debye-QNM description that is complementary to the standard QNM description. With this framework, we examine a scalar field propagating on two illustrative `Schwarzschild star' compact-body spacetimes: a neutron-star-like model $R>3M$ and an ultracompact object $R<3M$. We show that the closed-form Debye reconstruction matches well with the exact waveform, and that it converges even at early times, giving an accurate description of all waveform features including the prompt response. In the neutron-star case, the low-order Debye terms mainly describe the ringdown and a non-modal component associated with the sub-threshold branch cut. In the ultracompact case, the Debye series organizes the waveform into a prompt/ringdown contribution followed by a succession of echo-like wavepackets. The new Debye-QNM expansion and the standard QNM expansion have complementary spectral interpretations: the former identifies modes in individual propagation channels, whereas the latter describes collective resonances that are resummations of the former. This distinction clarifies how echo-like structures emerge from repeated interior propagation, and how pole and branch-cut contributions enter the time-domain signal.

We study the possibility of probing the dynamical inverse seesaw mechanism for the origin of light neutrino masses via the detection of stochastic gravitational waves (GW) in the low-frequency regime currently being probed by pulsar timing arrays. As the lepton number-violating term in inverse seesaw typically remains in the sub-MeV ballpark, its dynamical origin naturally brings the possibility of a low-scale first-order phase transition, which can be probed at low-frequency GW experiments. We also find interesting complementarity with heavy neutral lepton searches, as GW experiments remain sensitive to parameter space with small active-sterile mixing, which is out of reach for most particle physics experiments.

Condensable algebras in 2+1d non-chiral topological orders characterize gapped boundary conditions and interfaces. Applied to the Symmetry Topological Field Theory, they allow classification of symmetric gapped phases and impose sharp constraints on possible phase transitions. A condensable algebra is specified not only by its underlying set of anyons, which end on the boundary or interface, but also by its algebra structure. We introduce the concept of twin condensable algebras, which have the same anyon decomposition, but inequivalent algebra structure. We revisit the classification of condensable algebras in $\mathcal{Z}(\text{Vec}_G^\omega)$, i.e. in group-theoretical topological orders for finite groups $G$ with anomaly $\omega$. In this context we are able to identify twin algebras that arise from different mechanisms, such as subgroup data, SPT cocycles, and symmetry actions. In particular, we construct infinite families of examples of twins from so-called Gassmann triples, and exhibit cases in which the reduced topological orders are inequivalent despite having identical anyon content. Physically, twin algebras describe distinct symmetric phases that have isomorphic spaces of ground states, but inequivalent order parameters. Such twin phases never exhibit relative spontaneous symmetry breaking, and can be used to construct phase transitions without hidden symmetry breaking, which are intrinsically beyond Landau transitions.

We present exact analytic solutions for the linear dynamics of a two-field inflationary system in which the primordial curvature perturbation $\zeta$ is coupled to an isocurvature perturbation $\sigma$ of entropy mass $\mu$. The solutions are valid for arbitrary values of $\mu$ and the dimensionless interaction strength $\lambda$. They therefore provide analytic control over the strong mixing regime in which $\zeta$ interacts with light isocurvature fields, commonly associated with rapid-turn inflation. As a first application, we derive the amplitude of the primordial power spectrum in closed form, obtaining an expression that interpolates between the weakly mixed, strongly mixed, light-field, and heavy-field regimes. These results open the way to analytic studies of multifield observables beyond the power spectrum, including non-Gaussianity, particle production, and loop corrections.

We investigate the phase transition of Quantum Chromodynamics (QCD) with three degenerate quark flavors at zero baryon chemical potential. Using Möbius domain-wall fermions as the lattice fermion formulation, we ensure excellent chiral symmetry preservation. Our simulations are performed at three different temporal lattice extents, $N_{t}=6, 8, 12$, with a fixed lattice spacing $a=0.1361(20)$ fm, corresponding to temperatures of 242(4), 181(3), and 121(2) MeV, respectively. We explore a range of quark masses and spatial volumes with aspect ratios $N_{s}/N_{t}$ spanning from 2 to 4. By analyzing the mass and volume dependencies of the plaquette, plaquette susceptibility, chiral condensate, chiral susceptibilities, and Binder cumulant, we identify the pseudocritical transition quark masses from our largest lattice volumes. For $N_t=6$, this is 184(10) MeV (determined from the plaquette susceptibility). For $N_t=8$ and 12, the transition points vary slightly depending on whether the total or disconnected chiral susceptibility is used, yielding ranges of 36(1)-39.1(9) MeV and 3.5(3)-3.7(2) MeV, respectively, in the $\overline{\text{MS}}$ scheme at a scale of $\mu=2$ GeV. The negligible volume dependence at $N_t=6$ and 8, combined with finite-size scaling analysis at $N_t=12$ revealing volume growth significantly weaker than expected for a first- or second-order phase transition, points to a continuous crossover at these specific quark mass points. Additionally, we study the effects of residual chiral symmetry breaking on the chiral condensate and chiral susceptibilities using two different values of $L_s$.

The tidal deformability of a compact object is encoded in a single function of frequency, the retarded Green's function relating the induced multipole to the applied tide. Causality, reality, passivity, and the high-frequency conditions required for a positive-measure dispersion representation allow this response to be rescaled into a holomorphic self-map of the upper complex frequency half-plane. Using the Schwarz-Pick theorem, already employed to derive the quantum chaos bound in black hole physics, we provide a bound on the rate of the tidal response with respect to the frequency. For a neutron star with a positive tidal spectral threshold, the dynamical Love number is bounded in terms of the static response and the lowest frequency carrying nonzero tidal spectral weight, with the ideal single-mode model saturating the bound. For a black hole, whose tidal spectrum is gapless, the dynamical Love number is a finite-part moment of the horizon absorption spectrum. Under a growth hypothesis on the complete response the theorem constrains the full Euclidean response, but it does not bound the dynamical Love number.

This dissertation investigates scalar field descriptions of the dark sector (of the Universe), with emphasis on scalar-tensor cosmology, inflationary effective field theory, quintessence, and ekpyrotic/cyclic alternatives to inflation. The thesis first reviews scalar-tensor theories of gravity, starting from Brans-Dicke theory and extending to general non-minimally coupled scalar fields. Particular attention is paid to the interpretation of the scalar field, the relation between the Jordan and Einstein conformal frames, and the role of conformal transformations in cosmology. This provides the theoretical basis for treating scalar fields either as modified matter or as additional gravitational degrees of freedom. The second part discusses cosmological inflation from the viewpoint of effective field theory. The spontaneous breaking of time translations, the Goldstone description of adiabatic perturbations, higher-dimensional operators, radiative corrections and Planck-suppressed terms are analyzed in the context of ultraviolet sensitivity and the eta problem. The third part is devoted to quintessence as the simplest dynamical dark energy model. 'Tracking', 'scaling', 'thawing' and 'freezing' behaviors are discussed, together with observational diagnostics such as the statefinder parameters, the $(\omega_{\phi},\omega'_{\phi})$ plane and the $Om(z)$ diagnostic. The DESI DR2 results are used as motivation for considering time-dependent effective dark energy beyond a strict cosmological constant. The final part studies ekpyrotic and cyclic cosmologies. The ekpyrotic phase is analyzed as a smoothing mechanism based on an ultra-stiff scalar field equation of state. Perturbations, non-Gaussianity, tensor modes, bounce conditions, entropy production, geodesic incompleteness and phase-resolved field-space distance bounds are discussed.

Exceptional points (EPs) and eigenstate hysteresis, previously found in Kerr only with an auxiliary scalar field, arise intrinsically in the quasinormal-mode spectrum of Kerr-Newman black holes. Using a hyperboloidal formulation of the coupled gravito-electromagnetic Teukolsky equations and a Chebyshev pseudospectral method, we compute the $\ell=m=2$ spectrum and locate four EPs at up to $99.974\%$ of extremality, one from a new repulsion. The gaps close as a square root branch point, and quasistatic loops around one or more EPs exchange the coalescing modes, confirming hysteresis.